* Step 1: Sum WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X))
U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS)
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
afterNth(N,XS) -> snd(splitAt(N,XS))
and(tt(),X) -> activate(X)
fst(pair(X,Y)) -> X
head(cons(N,XS)) -> N
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
sel(N,XS) -> head(afterNth(N,XS))
snd(pair(X,Y)) -> Y
splitAt(0(),XS) -> pair(nil(),XS)
splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS))
tail(cons(N,XS)) -> activate(XS)
take(N,XS) -> fst(splitAt(N,XS))
- Signature:
{U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2} / {0/0
,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11,U12,activate,afterNth,and,fst,head,natsFrom,sel,snd
,splitAt,tail,take} and constructors {0,cons,n__natsFrom,nil,pair,s,tt}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X))
U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS)
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
afterNth(N,XS) -> snd(splitAt(N,XS))
and(tt(),X) -> activate(X)
fst(pair(X,Y)) -> X
head(cons(N,XS)) -> N
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
sel(N,XS) -> head(afterNth(N,XS))
snd(pair(X,Y)) -> Y
splitAt(0(),XS) -> pair(nil(),XS)
splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS))
tail(cons(N,XS)) -> activate(XS)
take(N,XS) -> fst(splitAt(N,XS))
- Signature:
{U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2} / {0/0
,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11,U12,activate,afterNth,and,fst,head,natsFrom,sel,snd
,splitAt,tail,take} and constructors {0,cons,n__natsFrom,nil,pair,s,tt}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X))
,splitAt#(activate(N),activate(XS))
,activate#(N)
,activate#(XS)
,activate#(X))
U12#(pair(YS,ZS),X) -> c_2(activate#(X))
activate#(X) -> c_3()
activate#(n__natsFrom(X)) -> c_4(natsFrom#(X))
afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS))
and#(tt(),X) -> c_6(activate#(X))
fst#(pair(X,Y)) -> c_7()
head#(cons(N,XS)) -> c_8()
natsFrom#(N) -> c_9()
natsFrom#(X) -> c_10()
sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS))
snd#(pair(X,Y)) -> c_12()
splitAt#(0(),XS) -> c_13()
splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS))
tail#(cons(N,XS)) -> c_15(activate#(XS))
take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS))
Weak DPs
and mark the set of starting terms.
* Step 3: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X))
,splitAt#(activate(N),activate(XS))
,activate#(N)
,activate#(XS)
,activate#(X))
U12#(pair(YS,ZS),X) -> c_2(activate#(X))
activate#(X) -> c_3()
activate#(n__natsFrom(X)) -> c_4(natsFrom#(X))
afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS))
and#(tt(),X) -> c_6(activate#(X))
fst#(pair(X,Y)) -> c_7()
head#(cons(N,XS)) -> c_8()
natsFrom#(N) -> c_9()
natsFrom#(X) -> c_10()
sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS))
snd#(pair(X,Y)) -> c_12()
splitAt#(0(),XS) -> c_13()
splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS))
tail#(cons(N,XS)) -> c_15(activate#(XS))
take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS))
- Weak TRS:
U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X))
U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS)
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
afterNth(N,XS) -> snd(splitAt(N,XS))
and(tt(),X) -> activate(X)
fst(pair(X,Y)) -> X
head(cons(N,XS)) -> N
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
sel(N,XS) -> head(afterNth(N,XS))
snd(pair(X,Y)) -> Y
splitAt(0(),XS) -> pair(nil(),XS)
splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS))
tail(cons(N,XS)) -> activate(XS)
take(N,XS) -> fst(splitAt(N,XS))
- Signature:
{U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4
,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1
,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/5,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0
,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/2,c_15/1,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#
,sel#,snd#,splitAt#,tail#,take#} and constructors {0,cons,n__natsFrom,nil,pair,s,tt}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X))
U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS)
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
afterNth(N,XS) -> snd(splitAt(N,XS))
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
snd(pair(X,Y)) -> Y
splitAt(0(),XS) -> pair(nil(),XS)
splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS))
U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X))
,splitAt#(activate(N),activate(XS))
,activate#(N)
,activate#(XS)
,activate#(X))
U12#(pair(YS,ZS),X) -> c_2(activate#(X))
activate#(X) -> c_3()
activate#(n__natsFrom(X)) -> c_4(natsFrom#(X))
afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS))
and#(tt(),X) -> c_6(activate#(X))
fst#(pair(X,Y)) -> c_7()
head#(cons(N,XS)) -> c_8()
natsFrom#(N) -> c_9()
natsFrom#(X) -> c_10()
sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS))
snd#(pair(X,Y)) -> c_12()
splitAt#(0(),XS) -> c_13()
splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS))
tail#(cons(N,XS)) -> c_15(activate#(XS))
take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS))
* Step 4: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X))
,splitAt#(activate(N),activate(XS))
,activate#(N)
,activate#(XS)
,activate#(X))
U12#(pair(YS,ZS),X) -> c_2(activate#(X))
activate#(X) -> c_3()
activate#(n__natsFrom(X)) -> c_4(natsFrom#(X))
afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS))
and#(tt(),X) -> c_6(activate#(X))
fst#(pair(X,Y)) -> c_7()
head#(cons(N,XS)) -> c_8()
natsFrom#(N) -> c_9()
natsFrom#(X) -> c_10()
sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS))
snd#(pair(X,Y)) -> c_12()
splitAt#(0(),XS) -> c_13()
splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS))
tail#(cons(N,XS)) -> c_15(activate#(XS))
take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS))
- Weak TRS:
U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X))
U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS)
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
afterNth(N,XS) -> snd(splitAt(N,XS))
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
snd(pair(X,Y)) -> Y
splitAt(0(),XS) -> pair(nil(),XS)
splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS))
- Signature:
{U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4
,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1
,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/5,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0
,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/2,c_15/1,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#
,sel#,snd#,splitAt#,tail#,take#} and constructors {0,cons,n__natsFrom,nil,pair,s,tt}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{3,7,8,9,10,12,13}
by application of
Pre({3,7,8,9,10,12,13}) = {1,2,4,5,6,11,14,15,16}.
Here rules are labelled as follows:
1: U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X))
,splitAt#(activate(N),activate(XS))
,activate#(N)
,activate#(XS)
,activate#(X))
2: U12#(pair(YS,ZS),X) -> c_2(activate#(X))
3: activate#(X) -> c_3()
4: activate#(n__natsFrom(X)) -> c_4(natsFrom#(X))
5: afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS))
6: and#(tt(),X) -> c_6(activate#(X))
7: fst#(pair(X,Y)) -> c_7()
8: head#(cons(N,XS)) -> c_8()
9: natsFrom#(N) -> c_9()
10: natsFrom#(X) -> c_10()
11: sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS))
12: snd#(pair(X,Y)) -> c_12()
13: splitAt#(0(),XS) -> c_13()
14: splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS))
15: tail#(cons(N,XS)) -> c_15(activate#(XS))
16: take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS))
* Step 5: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X))
,splitAt#(activate(N),activate(XS))
,activate#(N)
,activate#(XS)
,activate#(X))
U12#(pair(YS,ZS),X) -> c_2(activate#(X))
activate#(n__natsFrom(X)) -> c_4(natsFrom#(X))
afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS))
and#(tt(),X) -> c_6(activate#(X))
sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS))
splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS))
tail#(cons(N,XS)) -> c_15(activate#(XS))
take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS))
- Weak DPs:
activate#(X) -> c_3()
fst#(pair(X,Y)) -> c_7()
head#(cons(N,XS)) -> c_8()
natsFrom#(N) -> c_9()
natsFrom#(X) -> c_10()
snd#(pair(X,Y)) -> c_12()
splitAt#(0(),XS) -> c_13()
- Weak TRS:
U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X))
U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS)
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
afterNth(N,XS) -> snd(splitAt(N,XS))
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
snd(pair(X,Y)) -> Y
splitAt(0(),XS) -> pair(nil(),XS)
splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS))
- Signature:
{U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4
,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1
,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/5,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0
,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/2,c_15/1,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#
,sel#,snd#,splitAt#,tail#,take#} and constructors {0,cons,n__natsFrom,nil,pair,s,tt}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{3}
by application of
Pre({3}) = {1,2,5,7,8}.
Here rules are labelled as follows:
1: U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X))
,splitAt#(activate(N),activate(XS))
,activate#(N)
,activate#(XS)
,activate#(X))
2: U12#(pair(YS,ZS),X) -> c_2(activate#(X))
3: activate#(n__natsFrom(X)) -> c_4(natsFrom#(X))
4: afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS))
5: and#(tt(),X) -> c_6(activate#(X))
6: sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS))
7: splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS))
8: tail#(cons(N,XS)) -> c_15(activate#(XS))
9: take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS))
10: activate#(X) -> c_3()
11: fst#(pair(X,Y)) -> c_7()
12: head#(cons(N,XS)) -> c_8()
13: natsFrom#(N) -> c_9()
14: natsFrom#(X) -> c_10()
15: snd#(pair(X,Y)) -> c_12()
16: splitAt#(0(),XS) -> c_13()
* Step 6: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X))
,splitAt#(activate(N),activate(XS))
,activate#(N)
,activate#(XS)
,activate#(X))
U12#(pair(YS,ZS),X) -> c_2(activate#(X))
afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS))
and#(tt(),X) -> c_6(activate#(X))
sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS))
splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS))
tail#(cons(N,XS)) -> c_15(activate#(XS))
take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS))
- Weak DPs:
activate#(X) -> c_3()
activate#(n__natsFrom(X)) -> c_4(natsFrom#(X))
fst#(pair(X,Y)) -> c_7()
head#(cons(N,XS)) -> c_8()
natsFrom#(N) -> c_9()
natsFrom#(X) -> c_10()
snd#(pair(X,Y)) -> c_12()
splitAt#(0(),XS) -> c_13()
- Weak TRS:
U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X))
U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS)
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
afterNth(N,XS) -> snd(splitAt(N,XS))
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
snd(pair(X,Y)) -> Y
splitAt(0(),XS) -> pair(nil(),XS)
splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS))
- Signature:
{U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4
,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1
,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/5,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0
,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/2,c_15/1,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#
,sel#,snd#,splitAt#,tail#,take#} and constructors {0,cons,n__natsFrom,nil,pair,s,tt}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{2,4,7}
by application of
Pre({2,4,7}) = {1}.
Here rules are labelled as follows:
1: U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X))
,splitAt#(activate(N),activate(XS))
,activate#(N)
,activate#(XS)
,activate#(X))
2: U12#(pair(YS,ZS),X) -> c_2(activate#(X))
3: afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS))
4: and#(tt(),X) -> c_6(activate#(X))
5: sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS))
6: splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS))
7: tail#(cons(N,XS)) -> c_15(activate#(XS))
8: take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS))
9: activate#(X) -> c_3()
10: activate#(n__natsFrom(X)) -> c_4(natsFrom#(X))
11: fst#(pair(X,Y)) -> c_7()
12: head#(cons(N,XS)) -> c_8()
13: natsFrom#(N) -> c_9()
14: natsFrom#(X) -> c_10()
15: snd#(pair(X,Y)) -> c_12()
16: splitAt#(0(),XS) -> c_13()
* Step 7: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X))
,splitAt#(activate(N),activate(XS))
,activate#(N)
,activate#(XS)
,activate#(X))
afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS))
sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS))
splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS))
take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS))
- Weak DPs:
U12#(pair(YS,ZS),X) -> c_2(activate#(X))
activate#(X) -> c_3()
activate#(n__natsFrom(X)) -> c_4(natsFrom#(X))
and#(tt(),X) -> c_6(activate#(X))
fst#(pair(X,Y)) -> c_7()
head#(cons(N,XS)) -> c_8()
natsFrom#(N) -> c_9()
natsFrom#(X) -> c_10()
snd#(pair(X,Y)) -> c_12()
splitAt#(0(),XS) -> c_13()
tail#(cons(N,XS)) -> c_15(activate#(XS))
- Weak TRS:
U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X))
U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS)
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
afterNth(N,XS) -> snd(splitAt(N,XS))
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
snd(pair(X,Y)) -> Y
splitAt(0(),XS) -> pair(nil(),XS)
splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS))
- Signature:
{U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4
,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1
,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/5,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0
,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/2,c_15/1,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#
,sel#,snd#,splitAt#,tail#,take#} and constructors {0,cons,n__natsFrom,nil,pair,s,tt}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X))
,splitAt#(activate(N),activate(XS))
,activate#(N)
,activate#(XS)
,activate#(X))
-->_5 activate#(n__natsFrom(X)) -> c_4(natsFrom#(X)):8
-->_4 activate#(n__natsFrom(X)) -> c_4(natsFrom#(X)):8
-->_3 activate#(n__natsFrom(X)) -> c_4(natsFrom#(X)):8
-->_1 U12#(pair(YS,ZS),X) -> c_2(activate#(X)):6
-->_2 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)):4
-->_2 splitAt#(0(),XS) -> c_13():15
-->_5 activate#(X) -> c_3():7
-->_4 activate#(X) -> c_3():7
-->_3 activate#(X) -> c_3():7
2:S:afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS))
-->_2 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)):4
-->_2 splitAt#(0(),XS) -> c_13():15
-->_1 snd#(pair(X,Y)) -> c_12():14
3:S:sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS))
-->_1 head#(cons(N,XS)) -> c_8():11
-->_2 afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS)):2
4:S:splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS))
-->_2 activate#(n__natsFrom(X)) -> c_4(natsFrom#(X)):8
-->_2 activate#(X) -> c_3():7
-->_1 U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X))
,splitAt#(activate(N),activate(XS))
,activate#(N)
,activate#(XS)
,activate#(X)):1
5:S:take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS))
-->_2 splitAt#(0(),XS) -> c_13():15
-->_1 fst#(pair(X,Y)) -> c_7():10
-->_2 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)):4
6:W:U12#(pair(YS,ZS),X) -> c_2(activate#(X))
-->_1 activate#(n__natsFrom(X)) -> c_4(natsFrom#(X)):8
-->_1 activate#(X) -> c_3():7
7:W:activate#(X) -> c_3()
8:W:activate#(n__natsFrom(X)) -> c_4(natsFrom#(X))
-->_1 natsFrom#(X) -> c_10():13
-->_1 natsFrom#(N) -> c_9():12
9:W:and#(tt(),X) -> c_6(activate#(X))
-->_1 activate#(n__natsFrom(X)) -> c_4(natsFrom#(X)):8
-->_1 activate#(X) -> c_3():7
10:W:fst#(pair(X,Y)) -> c_7()
11:W:head#(cons(N,XS)) -> c_8()
12:W:natsFrom#(N) -> c_9()
13:W:natsFrom#(X) -> c_10()
14:W:snd#(pair(X,Y)) -> c_12()
15:W:splitAt#(0(),XS) -> c_13()
16:W:tail#(cons(N,XS)) -> c_15(activate#(XS))
-->_1 activate#(n__natsFrom(X)) -> c_4(natsFrom#(X)):8
-->_1 activate#(X) -> c_3():7
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
16: tail#(cons(N,XS)) -> c_15(activate#(XS))
9: and#(tt(),X) -> c_6(activate#(X))
10: fst#(pair(X,Y)) -> c_7()
11: head#(cons(N,XS)) -> c_8()
14: snd#(pair(X,Y)) -> c_12()
15: splitAt#(0(),XS) -> c_13()
6: U12#(pair(YS,ZS),X) -> c_2(activate#(X))
7: activate#(X) -> c_3()
8: activate#(n__natsFrom(X)) -> c_4(natsFrom#(X))
12: natsFrom#(N) -> c_9()
13: natsFrom#(X) -> c_10()
* Step 8: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X))
,splitAt#(activate(N),activate(XS))
,activate#(N)
,activate#(XS)
,activate#(X))
afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS))
sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS))
splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS))
take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS))
- Weak TRS:
U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X))
U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS)
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
afterNth(N,XS) -> snd(splitAt(N,XS))
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
snd(pair(X,Y)) -> Y
splitAt(0(),XS) -> pair(nil(),XS)
splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS))
- Signature:
{U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4
,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1
,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/5,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0
,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/2,c_15/1,c_16/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#
,sel#,snd#,splitAt#,tail#,take#} and constructors {0,cons,n__natsFrom,nil,pair,s,tt}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X))
,splitAt#(activate(N),activate(XS))
,activate#(N)
,activate#(XS)
,activate#(X))
-->_2 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)):4
2:S:afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS))
-->_2 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)):4
3:S:sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS))
-->_2 afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS)):2
4:S:splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS))
-->_1 U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X))
,splitAt#(activate(N),activate(XS))
,activate#(N)
,activate#(XS)
,activate#(X)):1
5:S:take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS))
-->_2 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)):4
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
afterNth#(N,XS) -> c_5(splitAt#(N,XS))
sel#(N,XS) -> c_11(afterNth#(N,XS))
splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
take#(N,XS) -> c_16(splitAt#(N,XS))
* Step 9: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
afterNth#(N,XS) -> c_5(splitAt#(N,XS))
sel#(N,XS) -> c_11(afterNth#(N,XS))
splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
take#(N,XS) -> c_16(splitAt#(N,XS))
- Weak TRS:
U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X))
U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS)
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
afterNth(N,XS) -> snd(splitAt(N,XS))
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
snd(pair(X,Y)) -> Y
splitAt(0(),XS) -> pair(nil(),XS)
splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS))
- Signature:
{U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4
,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1
,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0
,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#
,sel#,snd#,splitAt#,tail#,take#} and constructors {0,cons,n__natsFrom,nil,pair,s,tt}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
afterNth#(N,XS) -> c_5(splitAt#(N,XS))
sel#(N,XS) -> c_11(afterNth#(N,XS))
splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
take#(N,XS) -> c_16(splitAt#(N,XS))
* Step 10: RemoveHeads WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
afterNth#(N,XS) -> c_5(splitAt#(N,XS))
sel#(N,XS) -> c_11(afterNth#(N,XS))
splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
take#(N,XS) -> c_16(splitAt#(N,XS))
- Weak TRS:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
- Signature:
{U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4
,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1
,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0
,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#
,sel#,snd#,splitAt#,tail#,take#} and constructors {0,cons,n__natsFrom,nil,pair,s,tt}
+ Applied Processor:
RemoveHeads
+ Details:
Consider the dependency graph
1:S:U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
-->_1 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))):4
2:S:afterNth#(N,XS) -> c_5(splitAt#(N,XS))
-->_1 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))):4
3:S:sel#(N,XS) -> c_11(afterNth#(N,XS))
-->_1 afterNth#(N,XS) -> c_5(splitAt#(N,XS)):2
4:S:splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
-->_1 U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS))):1
5:S:take#(N,XS) -> c_16(splitAt#(N,XS))
-->_1 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))):4
Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
[(3,sel#(N,XS) -> c_11(afterNth#(N,XS))),(5,take#(N,XS) -> c_16(splitAt#(N,XS)))]
* Step 11: RemoveHeads WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
afterNth#(N,XS) -> c_5(splitAt#(N,XS))
splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
- Weak TRS:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
- Signature:
{U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4
,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1
,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0
,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#
,sel#,snd#,splitAt#,tail#,take#} and constructors {0,cons,n__natsFrom,nil,pair,s,tt}
+ Applied Processor:
RemoveHeads
+ Details:
Consider the dependency graph
1:S:U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
-->_1 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))):4
2:S:afterNth#(N,XS) -> c_5(splitAt#(N,XS))
-->_1 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))):4
4:S:splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
-->_1 U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS))):1
Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
[(2,afterNth#(N,XS) -> c_5(splitAt#(N,XS)))]
* Step 12: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
- Weak TRS:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
- Signature:
{U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4
,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1
,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0
,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#
,sel#,snd#,splitAt#,tail#,take#} and constructors {0,cons,n__natsFrom,nil,pair,s,tt}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
Consider the set of all dependency pairs
1: U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
4: splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
SPACE(?,?)on application of the dependency pairs
{1}
These cover all (indirect) predecessors of dependency pairs
{1,4}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
** Step 12.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
- Weak TRS:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
- Signature:
{U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4
,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1
,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0
,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#
,sel#,snd#,splitAt#,tail#,take#} and constructors {0,cons,n__natsFrom,nil,pair,s,tt}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_1) = {1},
uargs(c_14) = {1}
Following symbols are considered usable:
{activate,natsFrom,U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}
TcT has computed the following interpretation:
p(0) = [1]
p(U11) = [1] x1 + [1] x2 + [8]
p(U12) = [4]
p(activate) = [1] x1 + [0]
p(afterNth) = [1] x1 + [4]
p(and) = [4] x2 + [1]
p(cons) = [0]
p(fst) = [2]
p(head) = [2] x1 + [2]
p(n__natsFrom) = [7]
p(natsFrom) = [7]
p(nil) = [1]
p(pair) = [1] x1 + [1]
p(s) = [1] x1 + [2]
p(sel) = [2] x1 + [2] x2 + [1]
p(snd) = [2] x1 + [0]
p(splitAt) = [2] x1 + [1] x2 + [0]
p(tail) = [0]
p(take) = [4] x1 + [1] x2 + [1]
p(tt) = [1]
p(U11#) = [4] x1 + [8] x2 + [12]
p(U12#) = [1] x1 + [1] x2 + [1]
p(activate#) = [1] x1 + [2]
p(afterNth#) = [1] x1 + [0]
p(and#) = [1] x2 + [1]
p(fst#) = [2] x1 + [2]
p(head#) = [1] x1 + [1]
p(natsFrom#) = [1] x1 + [1]
p(sel#) = [1] x2 + [0]
p(snd#) = [1]
p(splitAt#) = [8] x1 + [0]
p(tail#) = [2] x1 + [0]
p(take#) = [1] x1 + [4] x2 + [4]
p(c_1) = [1] x1 + [15]
p(c_2) = [4] x1 + [2]
p(c_3) = [0]
p(c_4) = [1] x1 + [0]
p(c_5) = [0]
p(c_6) = [0]
p(c_7) = [8]
p(c_8) = [4]
p(c_9) = [1]
p(c_10) = [1]
p(c_11) = [1] x1 + [1]
p(c_12) = [1]
p(c_13) = [0]
p(c_14) = [1] x1 + [0]
p(c_15) = [2] x1 + [1]
p(c_16) = [8] x1 + [0]
Following rules are strictly oriented:
U11#(tt(),N,X,XS) = [8] N + [16]
> [8] N + [15]
= c_1(splitAt#(activate(N),activate(XS)))
Following rules are (at-least) weakly oriented:
splitAt#(s(N),cons(X,XS)) = [8] N + [16]
>= [8] N + [16]
= c_14(U11#(tt(),N,X,activate(XS)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__natsFrom(X)) = [7]
>= [7]
= natsFrom(X)
natsFrom(N) = [7]
>= [0]
= cons(N,n__natsFrom(s(N)))
natsFrom(X) = [7]
>= [7]
= n__natsFrom(X)
** Step 12.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
- Weak DPs:
U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
- Weak TRS:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
- Signature:
{U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4
,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1
,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0
,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#
,sel#,snd#,splitAt#,tail#,take#} and constructors {0,cons,n__natsFrom,nil,pair,s,tt}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
** Step 12.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
- Weak TRS:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
- Signature:
{U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4
,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1
,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0
,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#
,sel#,snd#,splitAt#,tail#,take#} and constructors {0,cons,n__natsFrom,nil,pair,s,tt}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
-->_1 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))):2
2:W:splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
-->_1 U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS))):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS)))
2: splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)))
** Step 12.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
activate(X) -> X
activate(n__natsFrom(X)) -> natsFrom(X)
natsFrom(N) -> cons(N,n__natsFrom(s(N)))
natsFrom(X) -> n__natsFrom(X)
- Signature:
{U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2,U11#/4
,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,sel#/2,snd#/1,splitAt#/2,tail#/1
,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0
,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#
,sel#,snd#,splitAt#,tail#,take#} and constructors {0,cons,n__natsFrom,nil,pair,s,tt}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))